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Doing the math for 738AP.

I tried doing the math and it didn't add up.
Factoring in talents and item bonus's and doing the equation in all possible ways:
result 1: 636.97
result 2: 641.90
(there were other results showing similar numbers)

Just to be clear:
-talents can provide +10 AP,+10.80AP(.60 per level) and +2%AP (bonus +2%AP at lvl 18).
-items alone are giving 473AP

So 473AP is the starting number and i added the bonus AP from ignite and archmage savvy because those are static values. Then in multiple different orders I added the % AP bonus's.
I couldn't figure out how this player was getting that amount of AP, it seems I come short just about 100AP almost every time. No Baron buff is or was present.

that's 2500 extra mp, plus his 1185 base, is 3685. Does Archangel's unique have a cap? I'm going to assume it's 600 like Tear, but I could be wrong. So 4285 mana.

You get a base 325 AP from the items, 78 from runes, 10.8 from masteries, 10 from Ignite, 428.5 from your archangel's, 65 from potion, 60 from baron. Veigar could theoretically gain 1 AP/4 seconds in a match, but let's say you're only doing 1/10 seconds on average given time running around and whatnot. After 30 minutes, that's 180 AP.